A partition u of [k]={1, 2,... , k} is contained in another partition v of [l] if [l] has a k-subset on which v induces u. We are interested in counting partitions v not containing a given partition u or a given set of partitions R. This concept is related to that of forbidden permutations. A strengthening of Stanley-Wilf conjecture is proposed.

We prove that the GF counting v is rational if (i) R is finite and the number of parts of v is fixed or if (ii) u has only singleton parts and at most one doubleton part. In fact, (ii) is an application of (i). As another application of (i) we prove that for each k the GF counting partitions with k pairs of crossing parts belongs to Z({1-4x}^{1/2}).